What does it mean for a country to have access to the markets. E.g: There is an ongoing discussion when Greece will follow a prosperity path in order to enter again the "markets".
When a country loses access to markets they mean access to debt markets. Here is a quote from a paper if you need a citation:
Greece lost access to credit markets when debt exceeded 130% of GDP, whereas Belgium successfully handled identical debt relative to GDP. What determines the highest debt that a country could repay, defined as “fiscal limit?” Can we estimate the fiscal limit for a country which has never lost access to markets? We use historical data on primary surpluses (surpluses excluding interest payments) and debt relative to GDP for six high-debt, developed countries, two of which lost access to markets after the 2008 global financial crisis, to estimate the behavior of the primary surplus in response to changes in debt. The high-debt countries include Belgium, Canada, France, Greece, Italy, and Portugal.
Predicting Sovereign Fiscal Crises: High-Debt Developed Countries (Daniel and Shiamptanis (2015))
Specifically, "closed" means that that the sovereign can no longer raise funds from capital markets. Depending on the specific situation, some entities within that country can continue to raise external funds. A business with significant assets or good prospects may be able to continue to raise funds even when the country cannot. In Greece, at least until recently, this was the case. See the article Greek companies get back on the road, with help from bonds.