To take multiple lines of input and put them in a list of strings, you can use sys.stdin.readlines() which will keep accepting returns until you give it an EOF [ctrl + d in *nix, ctrl + z in Windows].

E.g.:

user_input = sys.stdin.readlines("Favourite foods followed by EOF:") # User gives input... 
    print(user_input)
Output:
    >> ["Spam\n", "Spam\n", "Spam\n"]
Obviously this leaves you with newlines, so you could do something hacky like:
    user_input = [line.rstrip() for line in sys.stdin.readlines()]
Which will give you:
    >> ["Spam", "Spam", "Spam"]

Sounds like an interesting ML problem, though dealing with a list of strings may complicate things significantly. You might want to make them one long string using something like join, e.g.

" ".join(user_input) # Used with a separator of " ", but you could use newlines or something

Which gives one string of:
    >> "Spam Spam Spam"

Keep calling the input() function until the line it reads in is empty. The use the .split method (no args = space as deliminator). Use a list-comp to convert each string to an int and append this to your examList.

Here's the code:

examList = []
i = input()
while i != '':
    examList.append([int(s) for s in i.split()])
    i = input()

And with your input, examList is:

[[3], [2, 1], [1, 1, 0], [2, 1, 1], [4, 3, 0, 1, 2], [2], [1, 2], [1, 3]]

The above method is for Python3 which allows you to call input() and enter nothing (which is why we use this as a signal to check that we are done - i != '').

However, from the docs, we see that, in Python2, an empty entry to input() throws an EOF error. To get around this, I guess we could just make it so that the multi-line input is ended when the string such as: END is entered. This means that you must stop the entry with a line saying 'END':

examList = []
i = raw_input()
while i != 'END':
    examList.append([int(s) for s in i.split()])
    i = raw_input()

Note that I've used raw_input as to not perform type conversion.

which works the same as above.

raw_input can correctly handle the EOF, so we can write a loop, read till we have received an EOF (Ctrl-D) from user:

Python 3

print("Enter/Paste your content. Ctrl-D or Ctrl-Z ( windows ) to save it.")
contents = []
while True:
    try:
        line = input()
    except EOFError:
        break
    contents.append(line)

Python 2

print "Enter/Paste your content. Ctrl-D or Ctrl-Z ( windows ) to save it."
contents = []
while True:
    try:
        line = raw_input("")
    except EOFError:
        break
    contents.append(line)

If you just append the values from your initial input line to lst it allows you to take each line and make it into a 2D list

>>> lst = []
>>> lst.append([int(y) for y in input().split()])
5 2
>>> lst.append([int(y) for y in input().split()])
6 1
>>> lst
[[5, 2], [6, 1]]
>>> 

However, you mentioned multiple lines so I assume you want to take all the input at once. Using this answer, I adapted the code to append each line as a list (as above). Input will only stop when a Ctrl-D (posix) or Ctrl-Z (windows) character is found

>>> lst = []
>>> while True:
...     try:
...         line = input()
...     except EOFError:
...         break
...     lst.append([int(y) for y in line.split()])
... 
5 2
6 1
7 3
4 2
10 5
12 4
^Z
>>> lst
[[5, 2], [6, 1], [7, 3], [4, 2], [10, 5], [12, 4]]
>>> 

Hope this helps!

EDIT: Thought I would add, this could then be used later by iterating over the outer list

>>> for item in lst:
...     print(item)
... 
[5, 2]
[6, 1]
[7, 3]
[4, 2]
[10, 5]
[12, 4]
>>> 

Use a list comprehension, calling the input() function for each iteration, like so:

l = [int(input()) for _ in range(int(input()))]

output for print(l), with the first input being 4:

[1, 2, 3, 4]

The problem with your code is that input() stops as soon as you hit Enter, to get continuous input you need to use a either a while loop or a for loop and take input until user enters a sentinel value:

Using for-loop and iter:

def multiline_input(sentinel=''):
    for inp in iter(input, sentinel):
        yield inp.split()
...         
>>> lis = list(multiline_input())
1 2 3
400 500 600
a b c

>>> lis
[['1', '2', '3'], ['400', '500', '600'], ['a', 'b', 'c']]

Using while loop:

def multiline_input(sentinel=''):
    while True:
        inp = input()
        if inp != sentinel:
            yield inp.split()
        else:
            break
...             
>>> lis = list(multiline_input())
1 2
3 4 5
6 7 8 9
10

>>> lis
[['1', '2'], ['3', '4', '5'], ['6', '7', '8', '9'], ['10']]