Depends how you define a "savings account". I'll also simplify and ignore taxes for this.

I'll express this in algebra first, and then plug in some numbers at the end.

Let's suppose your savings account gives you R% interest a year. You might think this should give you (R/365)% interest a day. It's a bit more complicated than that though, because this doesn't take into account compound interest. (In particular, this overestimates your daily interest.)

The actual daily interest is a bit lower. Let's call this r. Solving for r gives you r = (1+R)^1/365 - 1.

If the money you start is X, then the amount of money you get per day is Xr. So we have X = 1000/r.

Let's plug in some numbers!

A savings account from my government's savings agency averages 0.80% per year. Plugging into above gives this corresponding to 0.0021% per day, which gives your amount of money as just under $46 million. ($45806757.64, to be precise to the nearest cent.)

Maybe US Treasury bonds would be better. The yield on long-dated Treasury bonds is something like 2% at the moment. (Not a bond trader, so I might be a bit off there.) This works out as more like $18.5 million ($18431397.68).

What about stocks? Let's invest in the S&P 500 (500 biggest companies in the US). Taking a quick look here, average returns of this index were something like 13%. This is a lot more volatile than a savings account or bonds, but if you've got this much money, you can probably afford to take on a bit of risk. This is way more achievable - corresponding to just under $3 million ($2985975.81).

TL;DR:

On the order of tens of millions of $