You can offer such services as a private person, but you will have to disclose your address.
First, let's discuss whether this is business activity that needs to be registered. A business activity (gewerbliche Tätigkeit) is any permanent/repeated activity that is intended to make a profit. Any repeated income can be evidence of business activity, unless it is one-off or e.g. from employment that is automatically taxed anyway. This means that e.g. running ads on your website, or asking for “donations”, even if they only pay a fraction of your hosting costs, could make it necessary to register a business. So could offering a free service with the intention to attract a later customer base.
You as a natural person have the right to enter contracts, and it is not necessary to create a legal entity/corporation. However, this also means that you are the contract partner, and are personally liable. Careful: a personal Rechtschutzversicherung might not cover disputes arising from running such a service.
§5 TMG establishes an imprint requirement (Impressumspflicht). You have to provide your name and the address where you live, and a quick electronic contact method such as an email address. This information must be easy to find.
This obligation applies to “geschäftsmäßige, in der Regel gegen Entgelt angebotene Telemedien” (roughly: business-like, usually paid tele-media). In practice nearly all German websites, internet services, and apps fall under this obligation. In contrast to gewerblich, geschäftsmäßig does not require a profit motive.