If you perform an in-place upgrade then no, the instance of SQL Server will be taken offline for the duration of the upgrade. I would not recommend this approach as it is quite risky as it can be difficult to rollback.

I would perform a side-by-side upgrade, meaning that I would build a new server, install SQL Server 2008 R2 (if you really have to go to 2008 R2, instead of 2012/2014) and then migrate the databases into the new server.

However there will still be downtime required when moving the databases.