The key to building such a reserve lies in your action of diligently setting aside whatever amount is deemed appropriate on a regular basis as well as in in the principle of the Time Value of Money. Yes, setting aside a budgeted amount each week will increase your reserve over time, but just as important, the time value of money works on increasing it the form of interest or other returns.
What is the Time Value of Money? The basic notion is that one dollar today is worth more than one dollar in the future. No, this is not actually due to inflation, but rather that your dollar set aside today will earn a percentage return if invested or placed somewhere other than beneath the mattress. If you set aside $100 now in a savings account that is earning 1% interest, then you will have $101 in one year. You may say that doesn't sound like much, so that is why finding an investment with a much better return is important. A modest return may be around 6%, meaning that $100 returns $6 and turns into $106 by the end of the year. Not much better? Well, that is why consistent saving is the key. Compound interest then begins to work in your favor. $100 put into that same account in the second year means that interest is earned on the $106 and the additional $100. That result is interest paid of $12.36. The fist year only paid you $6 and now the second year is paying you more than double that. I think you can probably start to see the pattern and apply it whether you are adding that $100 each month, each paycheck, or each week. The interest or return you get begins to build on itself rather quickly.
Many organizaitons and universities use such a principle when obtaining grants to fund certain campus projects. The funds required to construct a building, for example, are obtained and spent while an additional amount to maintain the building into perpetuity is obtained. That doesn't mean that infitnite dollars are being poured into the building, but that an amount is set aside to pay for the maintenance of the building. The amount set aside earns it's own income in the form of returns that exceeds the cost of maintaining the building. In other words, the money is doing the work. Individuals can also slowly build up such a reserve on the same principle.
The key to building your reserve is to not spend it. This may sound obvious, but many find themselves wanting that little extra cash for a new toy. Limit the reasons to tap into this reserve, and spend it for emergencies only. Don't spend the rainy day fund on a bright sunny warm day or you may regret it when the clouds start looming overhead.