Most modern Ethernet variants are designed for a bit error ratio (BER) of 10-12 or better. Early variants only had 10-10 or better, see IEEE 802.3 for details.
In term of packets/frames, it depends on their average size. A maximum sized, untagged Ethernet frame requires 1526 bytes or 12,208 bits on the wire, so a BER of 10-12 roughly translates to a frame error ratio of 1.2×10-7. That's worst case, within specs - with decent, not maxed-out cabling, you're usually quite a bit better, like 10-9 or so. The 4.83×10-3 frame error rate you've measured is very high - depending on the exact purpose of that cable you'd have to decide whether you can live with it or not.
However: if you fabricate your own cables you need to use a tester than can certify the cable ($1000+). Effectively, that Pockethernet tester of yours can only measure the error rate for a certain protocol. It doesn't measure attenuation over the frequency range, the crosstalk varieties or the various other parameters.
It's not enough to try-and-error because you might want to run other protocols on your cables later on - a Cat-5e DIY cable that's running 1000BASE-T fine might give up on 2.5GBASE-T since that uses a larger frequency envelope. Certified cables are guaranteed to work in all specified scenarios.