Q:

How low (and for how long) does the temperature need to be before I need to worry about covering my outdoor vegetable garden?

I have an outdoor vegetable garden in South Florida. The garden consists of things like tomatoes, peppers, lettuce, peas, and squash. Everything was started from seed about 2 months ago. I don't have any veggies yet but plenty of tall stems with leaves all over the place.
It generally never gets below freezing here, but we do have several days during the winter with relatively low overnight lows, ranging from high 30s to low 40s ºF (or 0–5 ºC). Daytime highs after such evenings are typically in the 50s and 60s ºF (or 10–20 ºC).
I'm told I should cover my garden with some plastic or a sheet in order to protect it from the cold. At what point is the temperature low enough (and how long does it have to be expected to be that low) before I need to take the time to do so?

A:

Different vegetables have different temperature thresholds.
Peas, for example, are very hardy -- they can tolerate heavy frost. (But not the blossoms -- if these get frosted you probably won't get any pods.) Kale and spinach can be planted in fall and survive months under several feet of snow snow to be eaten in the spring (if the deer don't eat them first).
Some plants can tolerate barely-freezing (30F), but die if temps drop to 20-25F. You might see these described as "semi hardy".
The tender vegetables like peppers, tomatoes, and squash won't tolerate any frost. If you're expecting frost you should cover those plants. Failing to do so will likely be fatal to the plants. A week of extended overnight temperatures of 35-40F may injure your plants to the point where they stop producing. If it's only 60F during the day, they certainly won't thrive. If you can manage to cover your plants and hold them around 50F overnight you may be able to keep them healthy enough to continue producing.
(I honestly don't have much experience with knowing "how long" tender plants can be exposed to cold and still make a recovery because once it starts to get cold here, it stays cold for long enough to kill all the tender stuff.)

A:

As @bstpierre says, the tolerances are going to vary. I've had peppers survive temperatures in the mid 30s but they go dormant and won't produce - a hard frost will kill them out right. I would expect the same from the closely related tomatoes. The downside is that your plants sound young so they might be more sensitive. As well as cloth, hay and leaves are another option to help keep them warm (I'm currently trying leaves wrapped in burlap/hessian on some banana tree stumps).
Also sensitivity does vary by variety and not just species. In our new house we moved the pepperocinis, chiltepins, and poinsettia peppers (our potted peppers from the old house) into a sheltered courtyard (North Texas). This probably still gets the hardest frosts, but they're against walls/windows of a heated house. The pepperocinis are dead, the chiltepins might be (or dormant - hard to tell at the moment!), and the poinsettias are very healthy! The best they've been since I've planted them. The poinsettias and chiltepins are now inside in an attempt to over-winter them.

