Ignore the fact that you may have continued to pay the Standing order. It is not relevant. You can easily argue that you paid by mistake. They cannot rely on your mistake to keep the money.
What is extremely relevant is whether you did, if you say, conclude the relationship, that you no longer needed the storage facility and whether that was conveyed to the storage company.
If they did not know you have removed all your belongings from them, then you are bound until you get noticed.
You say that you did handle all the financial details yourself and by that, I also assume that you mean the contract. However, someone did deal with it on your behalf with your authority. Hence, you are bound by that authority.
Need to look at the terms of the contract (if there is any written contract) with regard to how much notice you need to give to terminate the agreement and whether that can be by fax, email, verbally or was it must be in writing.
If there is nothing in the contract about that, then the notice that you give must be reasonable and it must be conveyed. If you simply took all your belongings out without saying whether you were coming or not, then it is highly likely that you would not be able to recover any money if they allege that they were keeping the facility open for you because you had not told them that you were vacated in.
However even if you had given notice and they were aware that you are vacationing, if they refuse to refund you you are faced with taking them to court to recover the cost which you have overpaid.
If there is any doubt as to whether you did give them notice to leave or not then once again, you are faced with court.
If you didn’t actually give them notice but they still supply what you empty all your stuff out completely, until the room was empty, you could argue that was “constructive notice” although they won’t refund, it’s a matter for the court to decide.
Can I clarify anything for you?
Please don’t forget to rate the service positive. It is an important part of the process by which experts get paid.