Read the synopsis and some reviews, it sounds pretty cringe. The whole idea behind the show is really far fetched though, REALLY far fetched. Space aliens? No problems. But this is trying to take something realistic and make it completely impossible. A space race like that would completely ruin both countries, Soviet Union especially. You have to realize that Soviet Union was a super-power militarily but its economic might was much smaller than that of the US. Today's Russia's economy is similar to Italy's. Soviet Union was a huge plot of land with nothing in it.
Show's not cringe at all - at least in my humble opinion and I tend to be a pain in the ass about suspension of disbelief. It is interesting - Season 1, set in the 1960s and 1970s, starts pretty close to reality and then deviates from there. Season 2, set in the 1980s, is full-on alternative history.
The initial premise - the Soviets beating the US to the moon - is actually far from far-fetched. As Von Braun and other NASA bigwigs later confessed, it'd likely would have happened if not for one or two design flaws in the mid-60s. I don't recall if that's actually the case, but I believe Korolev never died in the show. The following seasons, with nuclear-powered space shuttles (based on the real, never-realized Pathfinder project) are indeed a bit further from reality...l
I also caution against using GDP tables to measure the USSR's capabilities in regards to space, which fell under the general purview of the Soviet military, possibly only well-funded domain within the Soviet state - Soyuz used the same technology as ICBMs, which the Soviets were quite adept at producing. There's a reason why we (US of A) scraped the Space Shuttle and relied on the Soyuz in the 2010s to get our astronauts to space - it's a reliable and cost-effective launch platform.
You are dead right that their economy was... underwhelming. Up until the early 1980s, the USSR was the world's second largest gross national product (GNP - a metric some prefer when comparing command economies) but its economy was absolutely terrible at almost everything besides heavy industry. I'd be somewhat shocked if they were shown in the series to produce first-rate cars (as someone who drove a Lada...) or decent refrigerators, but rockets? That they were pretty good at and probably would have accomplished a lot more if they hadn't wound down the space race and diverted the budget to heavy armor and long-range missiles (both a colossal waste of money, but that's another conversation).
I'd give it a try. It's a pretty nice show.