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Romania mostly applies the same rules as the Schengen area, as part of their preparation to eventually join it. The rules detailed on the website of the Ministry of Foreign Affairs are nearly identical and even the forms or the stamps (technically not a visa) are almost the same as those of Schengen countries (including Estonia, as you can see on your own passport).
So the 90/180 rules applies and it's not possible to do a border run. If you have stayed 90 days (or thereabouts) in Romania and are not able to get a long-stay visa/permit, you will have to stay out of the country for another 90 days. However, because Romania is not part of the Schengen area, βout of the countryβ also includes Schengen countries.