How can one do this?
In my area, the rent alone takes away half of our paychecks. And we live in a very middle-class area.
The only way I could start saving in the first place was literally putting away $1 here and there.
I'd aim for a minimal savings of about $30 a month. This is approximately the amount for a 1 oz. American Eagle silver coin.
You'll have to cut expenses and expenditures.
You'll need to optimize your personal business to increase savings and reduce payments.
Example:
Pay off lower balance credit cards first. Right there would be $25 - $100 a month an individual could save from a portion. Take the remainder and pay off other cards. You'd be surprise how much you actually have when you aren't paying it out on credit cards. This is the "snowball method" recommended by Dave Ramsey.
Move closer to work. This saves money on gas and wear and tear on your car.
If you have credit, refinance your debt to consolidate and lower interest rates. This will net additional savings.
Clean up your credit report. The higher your score, the cheaper it is to finance credit.
Re-negotiate your insurances, cable/satellite, DSL, phone, etc. for greater savings, lower rates, while maintaining about the same service. In fact, I've lowered my payments and received greater benefits and services from providers while doing this.
Shop around for insurance. You'd be surprised what kind of savings you can obtain. Also review your policy for savings.
After achieving this increase in money savings, take that to begin stocking your "treasury".
This is from personal experience.