We would like to allow our clients and subcontractors to access the system so they could track the progress of work they are involved in. However, we don't want them to access the account's user directory. Is this possible?

Absolutely. The following access rights can be granted to each user regarding the "User Directory":  "Display All Users", "Display Only Administrators", "Users Assigned On The Same Projects", "Users In The Same Group" or "Disabled".