Buy a server HDD with lots of TBs for your data and use the SSD only for your OS.

It depends on how important your data is to you. Me personally, I just run a jellyfin server with dubiously acquired tv shows and movies. Which is to say that if I lost everything in a catastrophic failure, I wouldn't much care, so I decided to get a refurbished 14tb USB connected external HDD.

If you run anything more important, you should listen to others who might have a more robust solution

Add a second SSD, if the motherboard has a SATA port (I assume your current one is an NVMe drive). A SATA SSD is still more than fast enough as a second drive.

Moving to a bigger SSD also isn't too difficult, as long as you have a system where you can have both the old and new SSD connected at the same time. It can be a different system if needed. Download Clonezilla onto a USB stick with Ventoy on it, and boot into it. Just make sure you have backups and do the clone in the correct direction (don't clone the blank new drive onto the old one!!)

My low cost solution has been adding external mechanical disks. Those go up to several TB for cheap, so I put two and sync them with rsync weekly in case one suddenly fails.

As others have wisely said, keep the fast SSD for your OS, media rarely changes and is usually accessed sequentially, it can live in slower disks.

Probably the easiest solution would be to just chuck a larger disk in the system and retain the original drive for the operating system. If you do not need the high speed of an SSD, you may be able to get more storage space for the money by going with a spinning disk. 7200RPM drives are fast enough for most applications, though you may run into issues streaming 4K (or higher) resolution video.

Another option would be to start building out a storage pool using some type of RAID technology. On my own server, I use ZFS for the data partition. It is basically a software RAID. I use a RAID-Z1 configuration, which stripes the data over multiple disks (three in my case) and uses a parity calculation to provide data redundancy. It also has the advantage that it can be expanded to new disks dynamically and does not require that all disks are the same size. Initial setup does require more work and you are now monitoring multiple physical disks, but having a unified storage pool and redundancy is a nice way to go.

Any way you go, just make sure you have good backups. Drives fail, and sometimes even early in their life. Backblaze reports can be an interesting read when looking at drive options, as they really do put the drives through the wringer.