Atlassian have some information on the system requirements you can expect based on the issue count on your server: http://confluence.atlassian.com/display/JIRA/JIRA+Requirements#JIRARequirements-JIRAServerHardwareRecommendations
The article suggests that 1-2 GB should be the minimum for a small installation:
If you are planning to have a small
number of projects (10-20) with 1,000
to 5,000 issues in total and about
100-200 users, a recent server (2.0+
GHz quad core CPU) with 1-2 GB of
available RAM should cater for your
However they also state that their massive public Jira instance only requires 1.5 GB of RAM for Jira itself:
For reference, our http://jira.atlassian.com site has over 70,000 issues and over 30,000 user accounts. The system runs on a server with a 64-bit quad core processor and 32 GB of memory, of which only 1.5 GB is allocated to JIRA.
That would not take into account the other services running on the machine (e.g. a local database, OS needs)
Personally I run Jira with a local MySQL server on the same EC2 instance, but I use a 'small' instance. It does not fall within the free tier, but as I only turn it on as I need it, the cost is only a few dollars a month.