This claim exploits the coronavirus pandemic to push a recurring pro-Kremlin disinformation narrative about the EU's alleged imminent collapse. There is no evidence to support the doomsday claim that the pandemic is resulting in the death or obsolescence of the EU.
Although Ukraine is not part of the European Union, the country has signed an Association Agreement with the EU. Additionally, Ukrainian citizens were granted visa-free travel to the Schengen Area in 2017, for travel of up to 90 days within any 180-day period. While the business and leisure trips are both covered, Ukraine ETIAS does not include permission to work in the EU and a separate permit must be obtained, on which policies vary between Schengen member countries, and is open for everyone to apply.
Similarly, as part of the EU-Moldova Association Agreement, Moldovan citizens were granted visa-free travel to all 26 Schengen members. However, according to EU Regulation NO. 259/2014 citizens of the Republic of Moldova have to obtain a long-stay visa - Type D visa in order to work in the EU. Read a similar case here.
Regarding the current covoravirus crisis, and in order to cushion the blow to people’s livelihoods and the economy, the European Commission has adopted a comprehensive economic response to the outbreak, applied the full flexibility of the EU fiscal rules, has revised its State Aid rules and proposed to set up a €37 billion Coronavirus Response Investment Initiative to provide liquidity to small businesses and the health care sector.
Read more about the EU's response here and here.