The answer is YES. You can use solar energy to provide partial power to your home. You do not need any electricity storage at all to effectively use solar power on your home.
But, it doesn’t quite work like you describe in your question.
The majority of residential solar installations in the world are grid-tied. That means the solar inverter, which converts the DC electricity from the solar panels, to AC electricity usable by your home and appliances, is connected directly to the electrical grid. In these cases, the solar panels are not powering any individual appliances, they are supplying electricity to your whole household (typically through your main electrical service panel), and in the case where you are producing more solar electricity than your home is consuming, that excess electricity flows out onto the utility grid to be used by whatever loads out there can take it – typically your neighbors homes.
In the US, most utilities allow net-metering. In net-metering, you get credit for excess electricity produced during the day, that you can use to offset your consumption night, or any other time. Think of it as roll-over minutes for your electricity bill.
So, to answer your question, you could easily design a solar system that was sized appropriately to offset the usage of your server room. Again, you won’t be directly powering those devices, but the impact on your bill would be the same.
An important thing to note – in a grid-tied system, if the grid goes down (black out, brown out, etc), your solar stops working. Even during the day when the sun is shining.
If you want your solar to provide back-up power for this critical equipment in a power outage, then you likely will need a battery. It could be as simple as a UPS, as apposed to a battery integrated into your solar system.