Are there any databases that support protocol buffers?

Here is a project I found: https://github.com/google/mysql-protobuf

It's sql that speaks protobuf, looks promising but last commit was in mid March 2016.

ClickHouse lately added support for protobuf messages.

I am not aware of any (not that they don't exist, I just don't know about them), but perhaps it is worth discussing an alternate strategy and the pros and cons.

Typically you would want to store a serialized version of your protocol buffer, indexed by some particular key (a unique identifier, perhaps). You could then build secondary indexes for other interesting fields that point to that unique identifier. The idea is that you would want explicit indexes like this to scale beyond what a system that provides arbitrary indexes. This obviously opens up a world of new problems (such as stale arbitrary indexes).

ProfaneDB: http://profanedb.gitlab.io

"ProfaneDB is a gRPC interface between Protocol Buffers and RocksDB"

P.S: Though the question was asked 10 years ago I still find it very relevant, specially with the ascension of gRPC. Hope that can be of help.