That question is more involved than you think!
First of all: you're violating Nyquist. But, you're taking Nyquist's theorem for something that it is not.
Nyquist theorem, in essence says:
If we have a band-limited analog signal, we can faithfully represent it digitally (without losing any information) by sampling with a rate that is high enough.
So, first of all, from the info you're giving us, your signal is not band-limited. That should become obvious as soon as you realize that you draw the (theoretical) spectrum after the DAC: right, it's periodic with a period of 1 GHz, ie. the sampling rate. Period means it continues ad infinitum, i.e. it's not band-limited.
You don't say, so (,considering this is not your first question on this topic, so I presume that this means,) you don't have an anti-imaging analog filter after the DAC.
So, it'd be completely impossible to correctly sample this with a finite rate.
Then: you don't actually care about the analog signal, do you?
You want the bits. Good news: if you ADC samples exactly with the same rate as your DAC, then all the images will just alias back onto one, and everything will be fine.
Yeah. In theory.
In practice, that doesn't happen. First of all, and that is a very fundamental truth that I have to explain to a lot of people regularly, no two oscillators are exactly the same. Your system needs to have a bit of leeway for frequency deviation.
Then, all analog channels are frequency-dependent. That might also mean that they'll let the signal that you send and thus "imprint" on all these spectral repetitions through differently, depending on how that payload signal looks like in spectrum. Not to mention that it's rather rare that you can assume your channel stays like it is forever.
Then, you also get the problem of timing recovery. What use is your receiving ADC when it samples exactly in the middle between two transmit symbols?
So, you can do that, but you need at least timing recovery. And some DAC filtering (anti-image). And some noise and interference-limiting ADC filtering (anti-alias). And while you're at it, at least a simple equalization might be a good idea, and if it breaks down to an AGC.
Baseband BPSK signals (your signal is BPSK + an offset, if you will) aren't new. It's just that you typically avoid very much using them, unless your traces are very well-controlled (ie. you do 1Gb/s easily on a line between a RAM chip and a CPU that is routed on a PCB with proper impedance matching and plenty shielding, but you would avoid it very much on a cable with a plug and unknown length). For example, it's far easier to build Gigabit Ethernet, which transports 1Gb/s, too, using a signal that takes a lot of different states (not just two levels) but runs at 125 MHz, than to implement a 4Gb/s on/off scheme receiver for a fibreoptics cable (which really is just a DAC after a photo diode that either receives light – or doesn't).
So: high rate data is either pushed through very well-controlled channels in baseband (e.g. PCB traces, fibreoptics networking, twinax cable for SATA and 10GBase), or modulated onto a carrier and transported that way, or reduced in symbol rate by using high-bit-per-symbol mappings to limit the bandwidth and then transported in baseband (e.g. 1 Gb/s ethernet with 125 Mb) or both mapped to higher-order constellations and mixed onto a carrier (Digital TV, microwave links, LTE, basically all the radio you know that isn't low-rate).