What kind of Lisp system was used to run this? Would it work with any Common Lisp? What other things are required, like libraries or a database etc.?

I'm guessing reddit.asd lists some:

:tbnl (a toolkit for building dynamic websites)

:cl-ppcre (a regex library)

:trivial-http (a HTTP client)

:cl-who (a library for using S-expressions as HTML markup templates)

:clsql (a SQL database interface and ORM)

:clsql-postgresql (a PostgreSQL connector)

:cl-smtp (a SMTP client for sending email)

:ironclad (a cryptography library)