I don't think it's possible to do what you're suggesting with existing technology.
You have to remember, the sampling rate of your scope is very high (hundreds of MHz, if not GHz, for most decent scopes)
To continuously capture, the scope would have to be able to transmit the data to the computer as fast as it was capturing... and there's no communication method for the computer that can handle that. It would require something like connecting the scope's memory access bus directly to a dedicated bus on the computer so it could be communicated with directly by the processor... obviously a specialty application, which would require a completely custom setup... and don't forget, even a fast modern PC processor only operates at a few GHz, which wouldn't be nearly fast enough for fast scopes.
And even if you did get the data to the computer with such a direct memory access connection, you would still have no way to do anything with it! Computer memory operates at a few hundred MHz, and a hard drive is orders of magnitude slower. So you'd again be heavily limited on the scope sampling rate.
In other words, you'd probably have better luck building a new control board for the oscilloscope from scratch with more capture memory :lol:
Or just saving up the tens of thousands of dollars for a commercially available scope with large amounts of capture memory.
If you look here:
https://www.qsl.net/wd1v/scopefaq/open1.html
you'll see that some new scopes are handling up to 100 million sample points per capture, which equates to over 100 miles of consecutive scope screens if you were to scroll the entire waveform... should be enough for just about anything, I think