Buy Cash Flow $1000 At A Time - Passive?

Anything is possible, not everything is likely.

In order to cashflow $1000 with no mortgage, rent would likely need to be 1500ish to still cover taxes/insurance and random repairs.  Depending on where you buy, a home that rents for that price would likely be somewhere in the neighborhood of $150k-200k.  In some geographic regions you could do it cheaper than this, but I certainly wouldn't want to invest in those types of dying communities that are experiencing no population/wage growth, or worse yet negative growth.

At $1000/month cashflow, it would then take you between 16-23 years to save enough to buy your second property assuming 3% price appreciation and 3% rent increase per year.  (obviously having a high income job can reduce this time significantly).  

Without using loans and leveraging your money, it's going to be an uphill battle trying to achieve your goals unless you have a VERY high income job to help you buy a new property for cash every year or so.

If you assume a 150k purchase nets you 1k/month in cashflow, that is essentially an 8% return on your money.  At those ratios it would likely be better to simply put your money in a passive stock market index fund earning 10%ish over the long run.  Ya the real estate will likely come out ahead even at 8% once you factor in the homes appreciation, but if passivity is your goal and you don't want to manage or repair, then an index fund might better suit your needs.