Relatedly, there must be a (much longer) shortest sequence of bits that it is impossible to send over the internet, because beyond a certain length the payload

Relatedly, there must be a (much longer) shortest sequence of bits that it is impossible to send over the internet, because beyond a certain length the payload must be split over multiple packets, requiring additional header bits in between. In theory, the largest IPv4 packet that can be sent is 65,535 bytes. IPv6 is the same, but also allows for "jumbograms" that can be up to 4GB long, yeesh. However, the practical limit is the maximum frame size of the underlying link. (Standard) Ethernet and PPP both top out at 1500, ATM can handle 4456. Non-standard Ethernet jumbo frames can go up as high as 9216 bytes. But assuming you consider "send over the internet" to imply that most destinations on the open internet would be able to receive it, that means a frame size of 1500 bytes, giving you 12000 bits to play with. This leads to the curious result that it's impossible to send a sequence of more than 11870 "1"s. The shortest header you could have is a 160-bit IPv4 header, of which the last 32 bits (the destination address) can't all be 1 because 255.255.255.255 isn't routable, and actually 127.255.255.255 isn't either, so the earliest place you can start is bit 130. After 11870 "1"s, you reach the 4-bit version field of the next packet, which can't start with a 1 because it would indicate IPv8 or higher, and I don't think that exists... or at least it hasn't seen widespread adoption.