Carrots at the same temperature as my steak came out raw after two hours — why are vegetables completely different?

Two completely different mechanisms, and that is why more time did nothing.

Meat becomes tender because proteins change — they denature and connective tissue converts to gelatin. That starts happening at quite low temperatures, so a steak at 55 degrees is genuinely cooking, and a tough cut held for many hours at a low temperature becomes tender.

Vegetables become tender because pectin breaks down. Pectin is what glues plant cells together, and it barely degrades below about 83 degrees. Below that threshold you can hold a carrot for a day and it will stay crunchy, because the reaction you need is essentially not running.

That is the whole answer to your experiment: your bath was well below the temperature at which vegetables soften at all. Not slow — stopped.

So vegetables want 83 to 87 degrees, which is above the temperature you would ever cook a steak at, and roughly the temperature at which starch also gelatinises properly in potatoes.

Which answers your last question directly: no, not in one bath at one temperature. The two requirements do not overlap and no compromise temperature exists — anything hot enough for the carrots is well past well-done for the steak.

The good news is that vegetables at that temperature are quick. Carrots in something like an hour, depending on thickness. It is a sequencing problem, not an impossible one.

The practical way to run a whole meal with one machine is to use the temperature order to your advantage, because one direction is much easier than the other.

Vegetables first, high. Then drop the bath and do the meat. Getting a bath from 85 down to 55 is fast — add cold water and some ice and it is a couple of minutes. Going up takes much longer.

So the sequence:

1. Vegetables at 85, an hour or so. Bag them with butter and salt; they are sitting in whatever you put in there and it goes straight into them.
2. Pull them out and leave them in the bag. They hold heat well and they will be fine on the counter while you do the rest.
3. Cool the bath, put the meat in.
4. Sear the meat, and if the vegetables have cooled more than you want, they take thirty seconds in the same hot pan afterwards — which is an improvement rather than a rescue.

That last step is worth doing regardless. Sous vide vegetables come out evenly cooked and slightly flat, and a fast finish in a hot pan gives them the browning the water bath cannot.

If you cook this way often, a second cheap bath is the obvious answer, but the sequence above works with one and is what I do most weeks.