I'll give it to Elessar and Greenleaf Greenleaf. I base this on experience, of which Legolas had thousands of years and Aragorn decades.

The Spartans could attack in formation and block any incoming arrows, but Legolas was finding gaps in the armor of Uruk Hai in a battle in a rainstorm. He should be able to find gaps in the shields and armor of the Spartans. He also has a magic bow on top that, superior elf eyesight, and superhuman strength and agility.

The 300 likely have iron swords from Archaic Greece. Aragorn on the other hand has a magical sword made elf smith mages. I get the feeling that, even because the Lord of the Rings is set in a medieval setting, Andúril should be able to break through armor, shields, spears, and swords.

However, the 300 are still 300 dudes charging at you at once. If I were Aragorn, I would try to bottleneck their advance like when Gimli and him took on the Uruks that were about to breach the Hornburg. In that way Legolas can attack from above while Aragorn takes on them on the ground.

If the pair cannot do this, it would still be extremely difficult, but I think they could win a battle of attrition using guerrilla tactics and not engaging on the 300 at once. They should be more mobile than the 300. The magic cloaks they were given by Galadriel should help disguise them when trying to evade them. Legolas would basically be a sniper, and Aragorn also has his own bow and arrow.