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If your beans are bush variety, then you should put them each in their own medium-to-large pot. If they are climbing variety, you can plant them closer together (3-6" between plants) and let them climb up a panel of chicken/hog wire set into or next to the container(s). Here's a guide for spacing, based on the square-foot planting guide: http://journeytoforever.org/garden_space.html

Your beans can come out now for repotting, but leave the cukes in until their second set of leaves is well developed (1-2" in length). Dig down as far as you can on all sides to remove the seedlings, to avoid breaking the roots.

I think for cucumbers it's not the pot itself that's important but where you put it. They need to be able to climb and can get very big. They like sunny spots too. Compared to the aboveground portion of the plant their roots are small, so the pot(s) don't have to be gigantic.
Beans are not very fussy and imo you can leave them in that pot, if you move them around after removing the cukes, to give the bean plants some breathing room.

Hi Faith,

I think the safest thing would be to leave those beans right where they are. When I direct seed bush beans in the ground I only space them about 2 inches apart (and don't go back and thin them), so yours should be fine all in the one container. Plus I don't think beans take very well to transplanting.

As for the cucumbers, what they really need is space to trail above ground. I would very carefully dig them up (leaving lots of soil around the root system if you can so as not to disturb it too much) and put them each in their own pot.

In the future, you might want to keep some empty plastic containers (sour cream, yogurt, cottage cheese, etc.) on hand so if you run in into the same situation, you can just plunk each seedling into its own little pot.

I've been enjoying reading about your gardening adventures this spring - your excitement and enthusiasm is wonderful! : )

The cukes don't need to go in anything bigger than a coffee can. But you need to keep them well watered. Not enough water will make them bitter. Yuck.

I agree with the above commenters to leave the beans, but gently remove the cucumber seedlings into their own pots atleast 12"-14" in diameter. What is important is to stick a lattice in the pot in such a way so that the vine can climb up and trail above the dirt. I think a "tent" like trellis would be good. It must be sturdy.