Ideally there would be some way to convert EE image objects to sklearn-readable NumPy arrays directly using the EE Python API.
ee.Image.sampleRectangle() does this.
However, there is a limit of 262144 pixels that can be transferred. The interactive data transfer limit is in place to protect your system from hanging (it is easy to request terabytes of data without realizing it).
So in the case of a large area, your options are to export images to Google Drive or Google Cloud Storage and then import to Earth Engine Python API. Using Google Colab makes this easy - EE is installed by default and there is integration with GDrive and GCS. The Earth Engine batch task export methods are better equipped for dealing with large data (breaks up large exports into manageable sized GeoTIFFs).
ee.Image.sampleRectangle() may not be useful for your application, here is a demo in case it helps others.
The following Python script transfers three Landsat 8 bands for a rectangular region to the Python client and converts the EE arrays to numpy arrays and then stacks the arrays and displays the 3-D array as an RGB image representation of the region.
import numpy as np
import matplotlib.pyplot as plt
# Define an image.
img = ee.Image('LANDSAT/LC08/C01/T1_SR/LC08_038029_20180810') \
.select(['B4', 'B5', 'B6'])
# Define an area of interest.
aoi = ee.Geometry.Polygon(
[-110.6, 44.7]]], None, False)
# Get 2-d pixel array for AOI - returns feature with 2-D pixel array as property per band.
band_arrs = img.sampleRectangle(region=aoi)
# Get individual band arrays.
band_arr_b4 = band_arrs.get('B4')
band_arr_b5 = band_arrs.get('B5')
band_arr_b6 = band_arrs.get('B6')
# Transfer the arrays from server to client and cast as np array.
np_arr_b4 = np.array(band_arr_b4.getInfo())
np_arr_b5 = np.array(band_arr_b5.getInfo())
np_arr_b6 = np.array(band_arr_b6.getInfo())
# Expand the dimensions of the images so they can be concatenated into 3-D.
np_arr_b4 = np.expand_dims(np_arr_b4, 2)
np_arr_b5 = np.expand_dims(np_arr_b5, 2)
np_arr_b6 = np.expand_dims(np_arr_b6, 2)
# Stack the individual bands to make a 3-D array.
rgb_img = np.concatenate((np_arr_b6, np_arr_b5, np_arr_b4), 2)
# Scale the data to [0, 255] to show as an RGB image.
rgb_img_test = (255*((rgb_img - 100)/3500)).astype('uint8')