I'm looking for portable recording device that can record 4 pairs of left\right channels, total 8.
I found one that suits me, Zoom F8, but it is very expensive and almost not portable.
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I think you're asking too much here. The Zoom F8 is probably one of the cheapest 8 ch. portable solution available.
EDIT: Tascam made a "portastudio" too - the Tascam DP24 SD. It has 12 mono tracks and 6 stereo tracks, simultaneous recording of 8 tracks, simultaneous playback of 24 tracks and records on SD-Card.
You may be able to find an external audio interface which you can power up from a battery supply and use that along with a laptop or tablet of some sort. Two such interfaces are Focusrite Scarlett 18i8 and Behringer FCA1616, both running from an external 12V DC power supply which can be created easily with batteries. I'm not sure how "portable" this will end up being, but it will be cheaper, provided that you already have a laptop etc to use.
Last option: see if you can find the Zoom F8 used..
Depending on what features you require and the gear you have, the Zoom R16 might work. 8 inputs with preamps, records to an SD-card, 16-channel playback and runs on batteries if needed.
I don't know anything about the quality of the R16 compared to the F8, but it is a lot cheaper for the same number of inputs.
I'm afraid that you're probably not going to find anything much cheaper than a Zoom F8 if you're looking for an 8-channel field recorder. It should be perfectly portable though, if you buy an appropriate carry-case for it.
However, if you're limited by budget and need the recorder for a specific project then I'd highly recommend looking at renting one for a fraction of the price. Many film / audio equipment stores will also offer gear rental, as well as sales, so have a look around your local area and see what's available.
Depending on your budget and local availability, I'd recommend looking at renting a Sound Devices recorder, possibly a 788T. The best option though would be to go to / call your local gear rentals place and talk to them about your requirements, and they'll be able to set you up with a good package.
If you happen to be in London - UK, then I highly recommend Richmond Film Services. They have great rental prices, and will loan you all the cabling and peripherals at no extra cost.