Enforcing a no-fly zone would mean several things:
telling Russia that it can not fly its military aircraft over all, or portions, of Ukraine, a country that is not in NATO.
flying NATO aircraft over that airspace and using those NATO aircraft to shoot down Russian aircraft violating the no-fly order.
most likely it would also mean engaging with, and destroying, Russia surface to air missile systems (SAMs) that actively threaten NATO aircraft flying those interdiction patrols.
(the fact that the no-fly zone over Ukraine would be imposed at Ukraine's request, as an unjustly aggressed nation, changes zilch/nada/nothing to the above statements)
Further reading: THE DANGEROUS ALLURE OF THE NO-FLY ZONE
I am going to speculate here, but I suspect Zelinsky is well aware of the extreme nuclear escalation risks associated with this and only half-expects this to happen. However, he is also extremely media savvy and clearly understands that Western public opinion is distressed at "not doing enough" about Russia's aggression, especially as the war is progressing to a Grozny/Aleppo style of brutally leveling cities.
The fact that our leaders have no choice but to "meekly refuse to help" could be very helpful leverage to pressure Western countries to get more sanctions going against Russia as well as more lethal help.
Certainly while the unexpectedly dismal effectiveness of the Russian army is entirely due to Ukrainians' willingness to fight against overwhelming odds, the volume and severity of sanctions, boycotts and lethal aid against Russia - including from habitual foot-draggers like Germany - already far outweigh what anyone could have expected a month ago and may very well tip the balance in the long term. Maintaining the pressure on the West to keep up, and expand those sanctions is likely a key concern to Zelensky.
p.s. Previous no-fly zones, in Iraq and Yugoslavia, involved active shooting from NATO aircraft. There is no way around it unless the targeted nation submits from the onset.
p.p.s. No-fly zones aren't going to be enforced only by NATO surface to air missiles not located in Ukraine either. Even if that was technically possible, which I highly, highly, doubt (especially in a context where Russian planes being interdicted are flying low-level ground attack runs), that would just be inviting disaster like shooting down Russian medevac helicopters or the like. Additionally, it would only invite Russian to retaliate against units on NATO territory, not Ukraine, making this an even more daft policy. Last, Kiyv, the closest main fighting area is about 350km from Romania, the closest NATO country, while Odessa's area, closer to South Eastern Romania is still a considerable distance away (150k?) and sees no current fighting. Those are not within easy SAM ranges.
p.p.p.s As @llama points out, suppressing threatening Russian SAM batteries would be especially fraught for those based on Russian territory rather than in Ukraine itself.