Using Ensembl BioMART

1. Go to BioMART website.
2. Choose database: Ensembl Genes 70
3. Choose dataset: Homo sapiens
4. Click 'Attributes' then select the 'Sequences' option
5. Expand the sequences pane and select the 'Exon sequences' option
6. Expand the 'Header information' pane. Select the info you want to associate with each sequence record (e.g. Ensembl Gene ID, Ensembl Transcript ID, Ensembl Exon ID).
7. Click the 'Results' button.
8. Make sure the example output looks good, select the 'Compressed File' option and hit the 'Go' button.

The download took a while but eventually I got a file 'mart_export.txt.gz'. This file has output that looks like this

>ENSG00000264715|ENST00000578347|ENSE00002715610
AGGAGTGACCAGAAGACAAGAGTGCGAGCCTTCTGTTATGCCCAGACAGGGCCACCAGAGGGCTCCTTGGTCTAGTGGTAACGCCA 
>ENSG00000265161|ENST00000580394|ENSE00002716056
AGTAGAGATGGGGTTTCACCATGTTGGCCAGGCTGGTCTCAAACTCCTGACCTCAGGTGATCCATCCACC
>ENSG00000200917|ENST00000364047|ENSE00001438810
GCACATACATATACTAAAATTGGAACAATACAGAGAAGATTAGCATAGCCCCTGCGCAAGGATGACATGCAAATTCGTGAAGTGTTCCATATTAAA
...

get the coordinates:

Exon coordinates of hg19 genome download

and fetch the sequences:

How to get the sequence of a genomic region from UCSC?

Batch fetching fasta sequences from bed file

Have a look at the UCSC Table Browser: position base query

Here - ftp://ftp.ebi.ac.uk/pub/databases/astd/current_release/human/

If anyone is looking for the same but for GRCh37 here is the link.