It didn't
quite achieve them, but the Roman Empire of the 10th and 11th centuries very nearly reached its ideal natural borders from the perspective of the reduced Roman state of the 8th to mid-10th century. The Empire pushed to the Danube and secured the Branicevo-Nis corridor in Serbia, protecting it from the north; it secured the dependent status of the Dalmatian cities, Venice, and certain nearer Serbian principalities, protecting it from the north-west; it secured hegemony over Southern Italy, protecting it from the west; it reconquered Crete and reinforced its control over Cyprus, protecting it from the south; it conquered Cilicia, northern Syria (particularly the coast), and much of upper Mesopotamia, protecting it from the south-east; it conquered and bought (through princes bequeathing their realms to Rome in exchange for lands and titles) the better part of Armenia, protecting it from the east; and it secured control over Cherson and a few other outposts on the northern Black Sea, protecting it from the north and north-east. The only objectives remaining to the Roman government were to reconquer Sicily, a rich island of strategic importance with a large Roman population, and secure Tripoli in Syria, giving them full control over the Cypriot seaboard; it made a good attempt at the former, retaking half the island in the 1030's before being pushed back because of the leading general's rebellion after being replaced, and unsuccessfully besieged the latter a few times before a routinely renewed peace signed with the Fatimids in 1000 made the question moot. It's one of the great ironies of history that this apparently highly advantageous state of affairs barely lasted a few decades in its most complete form before beginning to collapse from immense unforeseen pressures on all sides, most notably from the Pechenegs, Turks, and Normans, the last two especially being extremely different from (and much more dangerous than) the traditional enemies faced by the Romans in Italy, the Balkans, and Anatolia.
The Roman Empire in 886 (maps my own, made in Google Earth):
The Roman Empire in 1045:
What these don't show, particularly in the latter (except in Venice), is a ring of client-states, protectorates, and associated principalities around the edges of proper imperial administration, particularly in the central Balkans, Southern Italy, Sardinia, Armenia, and the Kerch Strait, serving as friendly fortified outposts and relays for the projection of Roman soft power. As shown here, the Roman state in 886 encompassed 744,000 square kilometers of land, about the same as the HRE of 1000 AD, and 1,180,000 square kilometers in 1045, only slightly less than the Carolingian Empire in 810. The population of the former would have been around 9-10 million people, and the latter about 18-22 milllion, going by the estimates of Angeliki Laiou, centered on a Constantinople with a population of 150-200 and 300-400 thousand people respectively.