Hi Grace Bhang,
A "cramped" room/place/space is perfectly fine. It is idiomatic, and people do use it, so it will definitely get the point across. If the place you are referring to is "cramped" because there are many people (i.e. because it is crowded), you could also use adjectives like "crammed" (꽉 차있다), "packed/jam packed" (가득/꽉 차있다), "full", etc. "Tight" or "limited" also works but, personally, I'd just use "cramped". Literally translated your sentence goes, "(the place/space) it's too cramped, so you can't come in", or you can also just say, "there isn't enough space for you to come in; the place is too cramped/tight (limited in space)". Hope this helps.