When you 'own a bitcoin', what you actually own is a private key that can be used to spend the bitcoins that are located at the key's corresponding bitcoin address. Bitcoins themselves do not technically exist, instead the network simply keeps track of 'unspent outputs' at certain bitcoin addresses. Whoever owns a key that creates a valid signature to spend an unspent output can be said to "own the bitcoins at that address".
With that in mind, a wallet file is a collection of such private keys, which can be read by the wallet software so that you can utilize all of the wallet's functions.
However, it is possible to store just the private key itself in any format you desire. A private key is really just a very large random number, it can be encoded and stored in any number of ways. Keep in mind, if anyone is able to get a hold of the private key, they can steal your bitcoins. So it is a good idea to store the private keys in a very secure manner!
If you want to use the private key to spend your bitcoins (or perform any other function associated with a private key), then you would need to import that private key into some sort of wallet software.