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Consider using HP's OpenVMS, despite the name it's not open-source [or free] — but it does have a very, very good reputation for security. (Official Site)

If you're using DNS, there's now a fully formally verified DNS: Ironsides. — Full formal verification means that it has no buffer overflows, unintentional transfer-of-control, or unspecified termination.

As for web-servers, there's AWS¹ (Ada Web Server) or the Ade-AWA project².

¹ — I've only played with AWS, but it seems very stable; I haven't been able to get GnatColl [a project w/ DB support] to properly compile/link (I blame Cygwin) so I haven't been able to use that DB-interface.
² — The project seems tied to MySQL or SQLite DBs; I cannot in good consciousness recommend MySQL as a good DB. There also appears to be some limitation on OSes supported, even though Ada is pretty good at high-level platform-independence.