How long is a piece of string? 
It depends on the variety of tomato. You can get tiny tomato plants that grow no bigger than 6 inches, trailing tomatoes you can grow in a hanging basket or window box, or huge tomatoes that grow massive, tomatoes that crop all season or ones that all ripen at the same time. And, of course, there are all different types of tomatoes for different culinary uses...
They all need lots of sun though (at least 6 hours a day) so the first thing to do is identify the sunniest spots on your balcony and how much space you have in those places.
Based on my experience, I would go for small, bushy determinate varieties or the small trailing types - then you don't have to worry about them snapping in the wind, blowing over all the time and taking over all your outdoor space. They also don't need pruning like the indeterminate varieties. The trailing types are particularly good for balconies as you can grow them in hanging baskets or on window sills so they aren't taking up floor space or blocking light from the window (because they grow down), you also don't need to worry about getting your eye poked out by a cane... Maybe pick a few varieties that crop at different times. IME 2/3 small tomato plants per person that crop at different times will give you enough tomatoes for summer salads, unless you want loads for making sauces or snacking every day? I would also pick a disease resistant variety, there is nothing more heartbreaking than tending to them all summer only to have them get tomato blight.
It's too early to plant tomato seeds. You will need to wait until the end of March if you aren't growing them in a greenhouse so you have time to do some research.
If you pick a flavoursome variety, feed them and don't overwater them and it doesn't rain all summer, the taste is amazing... so intense, nothing like bland, watery supermarket tomatoes. It's really worth the effort for the taste! Ditto homegrown peas and strawberries. Cut and come again salad and herbs are also worth growing for the convenience/cost, although they don't taste much different from supermarket offerings.