All employees have a contract irrespective of whether anything exists in writing. (If you ask me to do your shopping for a fiver, and I agree, we've both entered into a legally-binding contract).
However your employer should have provided you with a 'written statement of employment particulars' within two months of the commencement of your employment. (Many employers ask you to sign to acknowledge receipt of the document but, just because you've not signed anything, it doesn't mean that you're not bound by any reasonable conditions of your employment).
If you choose to simple walk out of your job, the employer has the right to take legal action against you in order to recover any additional expenditure that your absence causes. For example, you might be employed for £6 per hour, for a 40-hour week. If the employer has to pay an agency £12 an hour to get someone to cover for you for the first 4 weeks after you leave (while the employer seeks another employee), you'll owe the employer £960.
However if the employer suffers no financial loss from your early absence, it's unlikely that he could take any effective action against you (except, of course, refusing to give you a reference).
Chris