This was a grand request of which I had little knowledge. It refers to the process during the 17th Century by which people's passage to the New World were paid in advance and then worked of by acting as a servant to the individual who paid the fees. This was the number 1 method of immigration from Europe during that period.
How does this apply to working for the Government? Well, you're looking at two different models. Working for the Government is much like a normal job, you're paid monthly and you have a task to perform. However, you are responsible and accountable to the people who voted for you. Thus you have a situation whereby an individual is effectively a servant to the people for a set contract - much like those with Indentured Servitude... However, that works on the assumption that the people rule the Government and not the other way around - which current models in the US and UK don't seem to match. But that's a whole other lengthy debate...
No relation. Military is a public service under which you (allegedly) provide protection for your nation (and their allies) against agressive/threatening forces and you recieve a pay check each month.
An indentured servant is someone who is attempting to work of his/her passage to the New World who works largely without pay until that amount is squared.