Q:

What happens if a Green Lantern is unable to recharge the ring at the end of the 24 hr time span?

If I remember correctly, each of the versions of Green Lantern summoned the power source lantern with their ring. Without power, the GL would die in a hostile atmosphere. How does a GL access the power lantern without a powered ring? If the GL is on the other side of the world, does he/she have to travel back to his/her residence to get to the power, or can a discharged ring still summon the power lantern? 

A:

Originally (even with the non-GL Corps Alan Scott stories) a Green Lantern had to charge their Ring every 24 hours. That, and a weakness to yellow objects, prevented the Corps Ring Slinger from being invincible. 
Often in the post-Alan Scott comic stories the Lantern of Sector 2814, Hal Jordan, would run out of charge mid-fight causing him to rely on his wits. In issue 46 of Green Lantern Vol 2, for example, the ring's energies expire and we see there's an emergency power reserve specifically to protect the bearer from mortal harm when the regular charge runs out.

I'm not sure if this answer is correct in the NEW 52! continuity but as Lantern & Bat Universes didn't change much after Flashpoint I expect the following to still be the case:
Currently the Green Lantern Power Ring acts like an electronic gadget and needs to be charged regularly. Now, though, the length of time is based on the charge not a specific time frame - the more a GL uses it, the faster its power runs out. Rings are capable of providing their bearer with its charge level upon request, they can be set to low-power-usage state or even be put into hibernation. Rings can also place its bearer into a protective bubble when its charge gets low, so that the Lantern might wait for rescue.
When the GL charges their Ring off of the personal lantern -personal power battery- it returns to the maximum charge. These batteries are directly linked to the Central Power Battery on Oa (the Lantern Corps Base of Operation) and themselves do not need recharging.
Jordan & Kyle Rayner used to kept their batteries nearby (at work or at home) for, if needed, they could travel at amazing speeds to retrieve them but that changed with the reforming of the Corps.

According to an entry on the DC Wiki for Green Lantern Power Battery - Batteries are often hidden in a pocket dimension when not in use. Which might indicate that a GL can access them at any time or any place. Yet, I could not find any specifics on this and as I've never seen a GL duck out of a fight on Alpha Centauri to recharge. As a result it seems that the bearer cannot access this pocket dimension from just anywhere.
Within the pages of the Green Lantern Corps regular series, which was launched off of the Green Lantern Corps: Recharge mini-series, several members of the Corps have run out of power or had their Rings drained during battle. As this was in the heat of combat they died.
According to an entry on ComicVine.com for Recharging, under Powers & Abilities - Other large sources of power may be used to recharge a Power Ring, however effectiveness may vary. There are many instances of this including when GL of 2814 fought the Old Doctor Light as well as when GL of 2814 teamed-up with the New Doctor Light, not to mention from proximity to Guardians (the founders and leaders of the GL Corps). 
It could be inferred that, were a Lantern in outer space a star might serve to recharge their Power Ring. However I could not find a specific instance of this.

A:

If a Green Lantern is in a hostile environment and unable to access a power battery, the most likely consequence will be: the Green Lantern ring wielder will die. Once a ring has completely drained its primary and secondary reserves, all of its defensive properties will lapse. Without power, the ring is just a piece of technology without power to perform any of its features.

This happens all too regularly doing the work in the peacekeeping operations Green Lanterns find themselves in. A ring will flash when only fifteen minutes of power remain. A ring also has a standing gauge so a ringbearer is always aware of how much power is available to them.

Rings have a primary power supply and a secondary in case they need to place the Green Lantern into a reserve power stasis. This is used to send the Lantern in the direction of potential rescue and await rescue. This low-power stasis protective state can last for months even though the ring is technically depleted.

If this too should fail, upon their death, the ring taps into a tertiary reserve and seeks out a new wearer.

The original Oan Power Rings required regular recharging. The time period was usually takes one rotation of the local planet or approximately 24 Earth hours, whichever came first.

The newer ring design (which was started by Ganthet for Kyle Rayner) now allows power rings to be charged and drained at a rate proportional to the energy expended by the wielder. With judicious use a ring's charge could last considerable longer than 24 hours.

A ring could be recharged:

By placing the ring in the Power Battery and holding it there for approximately ten seconds; the same time as required to recite the Green Lantern Oath.

By interacting with a Guardian, whose bodies are filled with a limitless supply of the Green Energy and can completely recharge any number of rings simultaneously.

A Guardian of Oa can even recharge a ring from a remote location by sending an energy twin to share the Green Power with a ringbearer.

The power ring can carry a charge for twenty-four hours before it is depleted. Minutes before it runs out of energy, the power ring will flash on and off, indicating its need for recharge at a power battery. Recharging takes approximately ten seconds. The ring can also be charged at the Central Power Battery and can draw off a charge from other rings. [Showcase #22, Tales of the Green Lantern Corps #1, GLS # 1]
Additionally, the Guardians of the Universe can recharge the power ring at will, as their bodies are storehouses of the emerald energies. Power can be drawn off their projections as well. [Tales of the Green Lantern Corps #3, GLGANTHET, GL3 # 48, DDAY ANNUAL # 1]

Lastly, a ring could be recharged by "bumming a charge" from another ring directly. This halves the energy of the both rings, reducing the time or charge to half full or 12 hours if the first ring was fully charged. This "splitting" of power also takes place if a newly charged ring is duplicated to recruit a new Green Lantern, or to provide aid to a Green Lantern whose ring has been destroyed or corrupted by an outside force.

Duplicated rings made from a primary Green Lantern cannot be used against their original ring, likely a safeguard. (Green Lantern (Vol 5) #5)

Regarding Power Batteries

Though commentary about Power Batteries says they can be stored in a pocket dimension, this appears to be apocryphal in nature. Most Green Lanterns store their lanterns on their homeworld base of operations. Given the extreme range of mobility the Oan Power Ring provides, a Power Battery is never more than a wormhole away from its user.

Power Batteries can cloak themselves from being visible and can be summoned to a Green Lantern after a period of time has elapsed as a safeguard. Sinestro was seen to do this in Green Lantern (Vol 5) #5. As to whether this is a common feature or something unique to Sinestro given his advanced skill using the Power Ring has not be determined.

Power batteries are durable but not indestructible, in a fashion similar to the Power Ring. If they are destroyed, they can create a destructive wave of force which varies in intensity capable of destroying a large city or a small Earth-size planet if the explosion is not controlled. A Green Lantern can both destroy their battery and recharge from it at the same time. They will not be able to recharge effectively unless they get a second battery.

On extended missions, Green Lanterns have been seen bearing their Power Battery with them if they are unsure they will be able to recharge at a designated facility in their duty area.

In the previous continuity the Green Power was related to and resembled the energy of the Source. Anyone capable of manipulating the Sources energy such as the New God's Highfather or Takion, is capable of controlling, draining or manipulating a Green Lantern's ring. Using the Source they could also recharge a ring in need of a charge.

