Being spiritual is, essentially, the expanding of awareness beyond the immediate towards the transcendent. Everybody does it, whether they hold materialistic worldviews or not.
Where most people get hung up is in this idea of belief. Belief is a powerful thing, it is essentially subordinating your will to something that only exists in your mind. Belief causes people to do amazing things, from the building of the pyramids to the creation of the Linux operating system.
You don’t have to have any beliefs in order to be spiritual. In fact, these days, the only way to arrive at any belief at all is to be spiritual. Keep chasing the transcendent until your personal truth manifests. It can touch mainstream religion or it can stay studiously away.