This is separate from the idea of an emergency or rainy day fund, which should be an additional 3-6 months expenses in an easy-to-access account, like a linked savings account.
I realize that if you're currently living paycheck-to-paycheck this can seem daunting. Definitely check out the video lessons on the YNAB site, and think about buying the software. There's probably a way to implement it in Quicken, but I have no idea.
https://www.youneedabudget.com/support/training-and-educatio...
Some might be turned off by having to manually input charges, by having to plan a budget each month, by being offended that they "could do the same thing in a weekend with Excel", but it's the best solution I've found for my wife and I. Not automated enough that we get disconnected from finances, but smart enough that we can easily see where we stand and run a few reports on our spending.
The system is pretty flexible, there's no need to categorize every stick of gum you buy, just use your best judgement and try not to cheat yourself.
What I like most about YNAB is the flexibility. It can be hard at times to follow the rules they set, but that is one of the rules! 'Roll with the punches'.
A paycheck every two weeks can never give you less than two paychecks a month. It'll give you three paychecks in two out of twelve months. So the issue you describe could never cause you to be unexpectedly short of cash? How could you have a "close call"? (I'm assuming here that a paycheck gives you money, and bills take it away.)
I think I'd like a service that let me get paid 1/14th of a paycheck every day and also pay [1/28 | 1/29 | 1/30 | 1/31] of my bills every day. Then, your day-to-day finances are literally an exercise of making sure there's always a small balance and that the pay-bill spread always remains positive.