Romania was once the Roman province of Dacia Traiana. As the Roman Empire collapsed and contracted, the province of Dacia Traiana was abandoned.

The locals, though abandoned by their government, retained their Roman language and culture. This romanized, Latin-speaking culture survived centuries of isolation and many waves of invaders from the steppe.

This is why the region is called 'Romania' - 'Land of the Romans'.

Incidentally, this is also why Romanian is so similar to classical Latin - not even French or Italian are as close.

so do the Romanians consider themselves Roman in anyway today

This is a tricky one. The level of perceived connection between Romans and Romanians fluctuated over the last 2 centuries, but one thing that remained constant was the linguistic connection, and our whole national identity is based around our "Latinity". As the saying goes, we are "an island of Latinity in a sea of Slavs".

Now regarding the above fluctuation: the "genetic" link (to use an anachronistic term) with the Romans changed over the years. In the initial stages of the development of Romanian nationalism (19th c.), the narrative was that the Romans conquered Dacia, slaughtered all the natives, and brought colonists from other parts of the empire, which over the centuries created the Romanian nation. This narrative was prominent out of a desire to be identified totally with a glorious past and glorious ancestors (the Roman Empire).

Over the decades however, the narrative shifted, as the Romanian state became more solidified and grew, and the provincial anxieties (being at the edge of Europe) gave way to anxieties about minorities (in the interwar period, a quarter of Greater Romania's population was non-Romanian). As such, the narrative shifted towards a more nativist view. The natives were not slaughtered, but were assimilated, and roman colonists were just a minority, albeit a culturally influential minority. Interest in Dacians started to grow, their position shifting from "barbarians worthy of extinction" to that of "our noble ancestors, full of patriotic dignity". This shift reached its peak in the 1980, when Dacocentrism overlapped with Ceauseascu's self-reliant ideology. See the wikiarticle on Protochronism for more detail.

Anyway, today's Romanians largely embrace the middle-of-the-road idea, where genetically we are descended from pre-Roman inhabitants, but culturally and especially linguistically we are descended from the Romans.

To a large extent, the cultural connection with the rest of Latinity has been exagerated, as a way to build a national narrative. We are culturally strongly connected with the peoples around us (Slavs and others from the Balkans) in a lot of ways (folk traditions, supertitions, cuisine, etc). The Roman Empire that had the most influence on us culturally was actually the Byzantine empire, and Nicolae Iorga called Wallachia and Moldova "Byzantium after Byzantium", even though we put much greater emphasis, on the Early imperial period - Trajan - because we speak a Latin-derived language (Byzantium was Greek-speaking) and because that's when Dacia was part of the Empire.

Source: History and Myth in Romanian Consciousness by Lucian Boia