Is it possible for humanity to leave the Milky Way Galaxy using only current technology and scientific knowledge?

 No, absolutely not!

Okay, wait maybe…

Wait again. NOOOOOOOO, it’s incredibly unlikely!

Let’s face the reality here. The edge of the Milky Way galaxy is about 25.000 light-years away. That is $2.37\cdot {10}^{17}$ kilometres. That is about 1.5 billion times the distance to our Sun. Even light needs 8,3 minutes to cover the distance to the Sun.

However, let's try to make a travel kit anyway after we realise that it probably takes more than 1 human life to get that mission done. Bring some toys.

Let's try to travel out of here:

If we travel as fast as the faster human-made object, the Voyager 2, we will have a travelling speed of about 54,000 km/h. Wooo, that's fast. Okay, it’s not actually that fast.

How long will we travel? Accounting for the fact that 25.000 light-years are equivalent to $2.37\cdot {10}^{17}km$ and the speed I mentioned above, we would only need a short time of

$t=\frac{distance}{velocity}=501,000,000$ years.

And now comes the fun part, which is why I said you might want to bring some toys. We need 501 million years to travel 25,000 light years, so we would need about 16.7 million generations if every generation is about 30 years apart.

So, bring lots of toys to your big ass space ship and you will have some babies who won't cry all the time. ;)