BUG: pandas 1.1.0 MemoryError using .astype("string") which worked using pandas 1.0.5
- [x] I have checked that this issue has not already been reported.

- [x] I have confirmed this bug exists on the latest version of pandas.

- [ ] (optional) I have confirmed this bug exists on the master branch of pandas.

---

#### Code Sample, a copy-pastable example

I tried to pinpoint the specific row which causes the error. The column has HTML data like this:

`
'''<div class="comment" dir="auto"><p dir="auto">Request <a href="/agent/tickets/test" target="_blank" rel="ticket">#test</a> "RE: **  M ..." Last comment in request </a>:</p> <p dir="auto"></p> <p dir="auto">Thank you</p> <p dir="auto">Stuff.</p> <p dir="auto">We will keep you posted .</p> <p dir="auto">Regards,</p> <p dir="auto">Name</p></div>'''`
 

```python

#This fails (memory error below):
df['event_html_body'].astype("string")

#Filtering the dataframe to only convert rows which are not null for this field **works**
x = df[~df.event_html_body.isnull()][['event_html_body']]
x['event_html_body'].astype("string")


#Filling NAs with another value fails:
df['event_html_body'].fillna('-').astype("string")
```

#### Problem description
I have code which has been converting columns to the "string" dtype and this has worked up until pandas 1.1.0

For example, I tried to process a file which I successfully processed in April and it works when I use .astype(str), but it fails when I use .astype("string") event though this worked in pandas 1.0.5.

The column does not **need** to be the new "string" type, but I wanted to raise this issue anyways.

Rows: 201368
Empty/null rows for the column in question: 189014 / 201368

So this column is quite sparse, and as I mentioned below if I filtered the nulls and then do .astype("string") then it will run fine. I am not sure why this worked before (same server, 64 GB of RAM), and this file was previous processed as a "string" before the update.

Error: 
```

MemoryError                               Traceback (most recent call last)
<ipython-input-38-939f88862e64> in <module>
----> 1 df['event_html_body'].astype("string")

/opt/conda/lib/python3.7/site-packages/pandas/core/generic.py in astype(self, dtype, copy, errors)
   5535         else:
   5536             # else, only a single dtype is given
-> 5537             new_data = self._mgr.astype(dtype=dtype, copy=copy, errors=errors,)
   5538             return self._constructor(new_data).__finalize__(self, method="astype")
   5539 

/opt/conda/lib/python3.7/site-packages/pandas/core/internals/managers.py in astype(self, dtype, copy, errors)
    565         self, dtype, copy: bool = False, errors: str = "raise"
    566     ) -> "BlockManager":
--> 567         return self.apply("astype", dtype=dtype, copy=copy, errors=errors)
    568 
    569     def convert(

/opt/conda/lib/python3.7/site-packages/pandas/core/internals/managers.py in apply(self, f, align_keys, **kwargs)
    394                 applied = b.apply(f, **kwargs)
    395             else:
--> 396                 applied = getattr(b, f)(**kwargs)
    397             result_blocks = _extend_blocks(applied, result_blocks)
    398 

/opt/conda/lib/python3.7/site-packages/pandas/core/internals/blocks.py in astype(self, dtype, copy, errors)
    588             vals1d = values.ravel()
    589             try:
--> 590                 values = astype_nansafe(vals1d, dtype, copy=True)
    591             except (ValueError, TypeError):
    592                 # e.g. astype_nansafe can fail on object-dtype of strings

/opt/conda/lib/python3.7/site-packages/pandas/core/dtypes/cast.py in astype_nansafe(arr, dtype, copy, skipna)
    911     # dispatch on extension dtype if needed
    912     if is_extension_array_dtype(dtype):
--> 913         return dtype.construct_array_type()._from_sequence(arr, dtype=dtype, copy=copy)
    914 
    915     if not isinstance(dtype, np.dtype):

/opt/conda/lib/python3.7/site-packages/pandas/core/arrays/string_.py in _from_sequence(cls, scalars, dtype, copy)
    215 
    216         # convert to str, then to object to avoid dtype like '<U3', then insert na_value
--> 217         result = np.asarray(result, dtype=str)
    218         result = np.asarray(result, dtype="object")
    219         if has_nans:

/opt/conda/lib/python3.7/site-packages/numpy/core/_asarray.py in asarray(a, dtype, order)
     83 
     84     """
---> 85     return array(a, dtype, copy=False, order=order)
     86 
     87 

MemoryError: Unable to allocate array with shape (201368,) and data type <U131880
```

#### Expected Output

#### Output of ``pd.show_versions()``

<details>


INSTALLED VERSIONS
------------------
commit           : d9fff2792bf16178d4e450fe7384244e50635733
python           : 3.7.4.final.0
python-bits      : 64
OS               : Linux
OS-release       : 5.3.0-1028-azure
Version          : #29~18.04.1-Ubuntu SMP Fri Jun 5 14:32:34 UTC 2020
machine          : x86_64
processor        : x86_64
byteorder        : little
LC_ALL           : en_US.UTF-8
LANG             : en_US.UTF-8
LOCALE           : en_US.UTF-8

pandas           : 1.1.0
numpy            : 1.17.5
pytz             : 2019.3
dateutil         : 2.8.1
pip              : 20.0.2
setuptools       : 45.2.0
Cython           : 0.29.14
pytest           : None
hypothesis       : None
sphinx           : None
blosc            : None
feather          : None
xlsxwriter       : None
lxml.etree       : None
html5lib         : None
pymysql          : None
psycopg2         : 2.8.4 (dt dec pq3 ext lo64)
jinja2           : 2.10.3
IPython          : 7.11.1
pandas_datareader: None
bs4              : 4.8.2
bottleneck       : None
fsspec           : 0.6.2
fastparquet      : None
gcsfs            : None
matplotlib       : 3.1.2
numexpr          : 2.7.1
odfpy            : None
openpyxl         : 3.0.4
pandas_gbq       : None
pyarrow          : 1.0.0
pytables         : None
pyxlsb           : 1.0.6
s3fs             : None
scipy            : 1.4.1
sqlalchemy       : 1.3.13
tables           : None
tabulate         : 0.8.6
xarray           : None
xlrd             : 1.2.0
xlwt             : None
numba            : 0.48.0


</details>

