I'm trying to make an elf barbarian and I'm having trouble figuring out her age, in human years she's 30 but I don't know if that would be the same in elven years (in the default 5e setting of the Forgotten Realms.) If so, would she be old enough to marry and go on an adventure?
Physically yes, up to a point - but that’s not all elves care about
While you can easily make this work for your game, the default assumptions about how elves age both physically and culturally are given in the Elf traits in chapter 2 of the Player’s Handbook (and the Basic Rules).
According to the "Age" trait, elves physically age (or at least develop) at the same rate as humans up until maturity. So a 30-year-old elf is physically an adult. But by elven standards, at 30 an elf is still a child in terms of experience, so your character might be considered too young for marriage and other adventures. From the book:
Age. Although elves reach physical maturity at about the same age as humans, the elven understanding of adulthood goes beyond physical growth to encompass worldly experience. An elf typically claims adulthood and an adult name around the age of 100 and can live to be 750 years old.
After reaching physical maturity, it’s less clear how elf years compare to human ones. Since they live much longer than humans, they have a different perspective on time, which is explained mainly in personality terms in the rest of the Elf description. While it’s not stated, it’s clear elves physically age much more slowly than humans - if at all - once they reach adulthood, in order to live for 700 years or more. No specifics are given about this, though, so you are free to interpret what that means for older elves however you like. It is common to assume that like Tolkien’s elves, they do not physically age once they become adults, but you can decide they just age very slowly instead.
Of note: 18 years is a little less than a quarter of the expected (D&D) human lifespan, but 100 years is less than one-seventh of a typical elf lifespan. So even if a child for 100 years, an elf spends a much larger portion of their life as an adult, which is worth taking into account when considering your character.
It depends on her background. If she were raised in a typical elven society then she is too young. As Guybrush McKenzie answered, elves reach physical maturity at the same age as humans, but socially they are considered adults at around the age of 100. But even if she were raised by elves she could be an exception to the rules. Circumstances could have forced her into adventuring life way earlier than she was ready for, and forced her to grow up faster. Maybe her settlement was attacked by raiders, or her kingdom was involved in a war, or she was kidnaped,...
But if she was raised by humans (or other short-lived people) then she is old enough. Physically she was grown up at 15-20, and since everyone in human society is considered an adult by then it would be completely normal for her to be married at 30. Example of such elf is Dahlia Syn'dalay, who entered into the service of Szass Tam of Thay (one of the rulers of Thay) in her 20s.