I don't recommend using pots without drain holes. However, my work sometimes requires me to do just that and it is possible to successfully maintain plants that way. If you are reasonably experienced with plants and have a pretty good idea how much water plants tend to use, then it should not be feared.
As long as you don't add water until the soil has dried to an appropriate depth (surface for some plants, halfway down for others) and it reaches that level of dryness every week or two, then you should be okay. Make sure you don't use hard water or much fertilizer as the excess mineral salts have no way to flush through and will accumulate and cause root damage.
Fear not, but I repeat it is always preferable to use pots with drain holes!
