how long can i get them to last?

 

how long can i get them to last?

It depends on how much current you require over what period of time and what minimum voltage will determine when the battery gets replaced.

3000 mAh /50μA = 60,000 h = 6.85 yr.
If at the end of that amount of time the battery was still able to produce 3.0V that would be a pretty good deal.

OTOH
3000 mAh/80 mA = 37.5 h, which sounds like you might need to change the battery more frequently. Between those two extremes there must be a compromise figure.

OK
3000 mAh / 4.4 mA ≈ 682 h ≈ 4 weeks. That sounds about right, change the battery once a month.

 

how long can i get them to last?

How often will you operate the relay, and what current does the relay coil require.

 

how long can i get them to last?

You should come at it from the other direction -- how long do you NEED them to last?

What kind of batteries are you using? Rechargeable? Alkaline (or other primary cell)?

If someone designed the system for you and said that you would need to replace (or recharge) the batteries daily, would that be acceptable? If not, what is the shortest amount of time between replacements/recharges that would grudgingly be considered tolerable? That will give you a place to start.

Next, you can either design the circuit with an eye toward minimizing the current and then, once designed (at least enough to get current draw estimates), you can determine what the battery capacity is that you need and then see if batteries with at least the capacity (as well as accepted voltage, current, and other ratings, of course) are acceptable in terms of cost, size, weight, or whatever else is important.

The other way is to do the opposite -- pick the largest battery that is acceptable in terms of cost, size, and weight and, from there, determine the current budget you have to stay within and design the circuit accordingly.

Which of these approaches is best depends on the problem (which hasn't been spelled out nearly well enough to get specific recommendations) and what is truly important that any acceptable solution must satisfy.