Most browsers support cookies of up to 4096 bytes. That is plenty of space for storing a few values on the user's computers, but you would not want to try to store a dataset or some other potentially large piece of data in a cookie. More practically, you probably do not want to store a big collection of user information in a cookie. Instead, you would want to store a user number or other identifier. Then, when the user visits your site again, you would use the user ID to look up user details in a database.
Browsers also impose limitations on how many cookies your site can store on the user's computer. Most browsers allow only 20 cookies per site; if you try to store more, the oldest cookies are discarded. Some browsers also put an absolute limit, usually 300, on the number of cookies they will accept from all sites combined.
A cookie limitation that you are likelier to run into is that users can set their browser to refuse cookies. You cannot do much to get around this problem except to avoid cookies altogether and use a different mechanism to store user-specific information. A common method for storing user information is Session state, but Session state depends on cookies, as I explain later in Cookies and Session.