Yes you can. Create a group, add the users you want access to the project, create a permissions schema that allows them access and make sure that not the rest of the projects on the JIRA instance allows anonymous access, as this would grant them access to. The simplest way to see if some projects can be browsed by anyone is by simply logging out from JIRA and then visit the dashboard.
Amy, Noam and Jonas are correct, but let me see if I can clear things up for you.
Hope this helps.
Create a new Permission scheme:
And add the group of users to the new permission scheme Browse Projects and take them out of the old permission scheme Browse Projects permission.
Now configure the relevant project that you what then to see to the new permission scheme.
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