Is it impossible to speak a second language as perfectly as native speakers? Is it really important to make myself sound like the native speakers when I speak English or Spanish? How do I completely get rid of the accents?

• the less common uses of ain't  (He ain't been around since last year)
• the use of the auxiliary do with got (Do you got any quarters?  I ain't got nothin', do I?). 
• There are contractions that aren't really taught in schools (I'm'n'a head on down there later).
• Foreign speakers use so-called phrasal verbs far less frequently than native speakers do, because they're really difficult.  They'll use common phrasal verbs (sit down, throw up), but instead of less common ones (take off meaning leave fast or sit around meaning lazily hang out), they tend to use single-word forms or adverbs.  Idiomatic ones, they'll replace with common ones (Everything will calm down instead of It'll all blow over.)
• And there are complex forms that second language learners don't use much, like tag questions (We're not going, are we?)
• or auxiliary dropping (Got any quarters?)
• and of course, there are idioms out the wazoo. 

Impossible, no. But this all depends on your affinity for languages. Andrew broke this down beautifully, but I will add my own personal experience here:

My mom’s first language is Spanish. She learned English in school, right around Kindergarten. For the most part, she has a typical New York (more specifically, Manhattan) accent, but some words give her away as a native Spanish speaker. In Spanish, the placement of the tongue when pronouncing “L” is behind the teeth, closer to the roof of the mouth - in English it tends to be between the teeth, but depending on the region/dialect of English that is spoken, it can also be behind the teeth, just closer to them than the roof of the mouth, if that makes sense. She is fully fluent in both languages, though, and her pronunciation is impeccable. She also learned French in college and though she's lost most of it, she reads and understands it and pronounces it well (albeit not like a native French speaker).

My first language is English. I learned Spanish at home around the age of 5/6, but spoke mostly English both at home and in school. I didn't really start caring about Spanish until my late teens/early 20s, and didn't start speaking it fully until around age 25 (I’ll be 34 next month). Today I am fluent, but make small grammatical mistakes that mom, who is a grammar Nazi in English AND Spanish, loves to correct me on. I've been told by my native Puerto Rican family members who were born on the island that I sound like them and that they would never know English was my first language. But this is largely because I was exposed to Spanish since a baby and I know what it's supposed to sound like, if that makes sense. My Spanish accent mirrors hers as a native speaker and my English accent mirrors my dad’s, who is from Brooklyn :)

My fiancé is Croatian American. He was born here, but his first language is Croatian. He learned English in school and is fully fluent in both languages. He has a very, very slight accent (he’ll deny it!) on certain words that give him away as a non-native English speaker. His mom, who speaks English, Croatian and Romanian, has a heavier accent, but is still very much American since she came here at fourteen. But I notice that his pronunciation of certain words mirrors hers. His Croatian accent mirrors his dad’s, who speaks mostly Croatian and a little English. (Interestingly, his younger brother does not speak Croatian at all - only English. He does understand it, though.)

In all three cases, these individuals had several things going for them that helped with their accents:

1) They were exposed to the language from an early age (important, but not a deal breaker if this is not the case for you)

2) They each have a natural affinity for language and are able to pick things up quickly (more important, although anyone can get good at something if they work hard enough at it)

3) They had native speakers to practice with regularly and get the flow/nuances of the language that is only possible with exposure to native speakers (in my view, this is most important)

So, to your original question of how to completely get rid of the accent….there are accent reduction classes out there that actors take to reduce their accents and sound more neutral. But if you don't want to go that route, the best advice I can give you is practice as much as possible with a native speaker and work diligently at pronouncing things as they do. Hope that helps!

You cannot really get rid of your accent. In fact, every English speaker has some "accent" (dialect, etc.) that is their own. I say embrace your accent. You speak perfectly fine, from what I've gathered. If people mock your accent, tell them you speak three languages. That usually shuts their mouth (I am trilingual myself, so I know how it feels.)
My first language was a combination of all three that I was exposed to: Spanish by my mother, Farsi by my father, and English from both and the outside world. I have an accent in all three languages, and I get complaints about it. My mother's side wishes I spoke Spanish perfectly, my father's wants my Farsi perfect, and my classmates my English. Once I tell them I speak three, they feel flabbergasted and drop the topic of my accent completely. Overall, just embrace your accent- it is a part of you.

I am Danish, and Danish is my first language. For over two years, I was in a relationship with a Brit. I was what you'd call fluent in English before the relationship, but of course the daily interactions had an effect on me.

I obviously met a lot of my ex-partner's friends, and many were surprised, shocked even, to discover that English wasn't my first language. So yes, I'd say it is possible to speak a second language as well as the native speakers, but it does depend on you language skills. Some people just pick it up easilier than others. Also, I have to remind you, that if you study a language theoretically rather than practically, you will end up with a better grasp on the language than the native speakers, which will make your second language unauthentic.

However, adopting a "second first language" comes with a price. Long before I got into this relationship, when I was about a semester into an international education, my brother told me, out of the blue, that I had lost my Danish accent.

Whether it is important or not depends on what you need your second language for. All in all, I wouldn't say that having a second first language is important. It's impressive, but it's not a skill you "need". As long as people understand what you're trying to say, it doesn't matter if you say it with an accent.

Hi! I'd like to say it is, but that it's extremely difficult as other people say.

I am a bilingual but then I grew up with reading thousands of English books and watching cartoons every other day. It's widely known that we lose the ability to pronounce phonemes in our non-native language when we are children. After adolescence, removing accents completely from a second(or a third, fourth...) are very difficult and can only be done with extensive practice over years. One way of lessening foreign accents I found while tutoring people would be to listen to high-quality lectures/speeches (i.e. TED) and trying to repeat what the speaker says, accent, rhythm and all.

But as a functioning but glaringly non-native french speaker, I'd like to say the perfect accent doesn't matter as much as you may believe - you seem to articulate yourself in English very well, why do you feel that you have to conform yourself to that one, specific type of American English? After all, English is a staggeringly global language!

As long as you get yourself across, and speak with confidence, I believe people who mock how you speak are jerks and should be ignored.