Is it possible for me to allow other Google users to view my contacts in Google Contacts, or perhaps only certain groups of contacts? Or is this a feature I'd need Google Apps for?
With the delegate option that's possible:
As Jacob mentioned, to my knowledge, the only way to currently do this is via Google Apps accounts & Delegation. I don't think a way exists to do it with a standard account.
Your best bet might be to look for a CRM that syncs with Google Contacts - it might be a little overkill for your needs (depending on the context), but it might work. However, in my research it seems like a good number of the CRMs are based on Google Apps accounts only, so that could be a roadblock. But worth looking into further.
We're currently trying to solve this problem at FullContact (yes, I work for them, so take my recommendation with a grain of salt - I'm biased). Our cloud address book pulls in Google Contacts and lets you organize and create shared lists of contacts, then syncs directly back to Google Contacts. We'll begin releasing beta invites in the near future - but unfortunately, it doesn't solve your problem now.
At the moment, since I'm not aware of any add-on app that offers this functionality, the easiest path to your result would probably be to just upgrade to a Google Apps account and call it good.