I’m not up-to-date with current NAS systems anymore – I’m running an older QNAP NAS (TS-453), and it has their proprietary “Container Station” which can run web applications in Docker + LXD containers. Not FOSS, though the containers very much are and can be moved to other systems.
As an alternative, FreeNAS/TrueNAS sells NAS systems where at least the software side is FOSS. They’re quite expensive, though.
The prices of other brands also quickly breach silly levels, but a basic 2-bay NAS is about ~$250 for QNAP, ~$200 for Synology and ~$1000 for a TrueNAS. Without hard drives.
If you’re not interested in the data storage side, a Mini PC w/Proxmox (popular Docker/LXD container engine w/browser-based management) or even a direct install on a Raspberry PI are possible for under $100.