Short answer: No.
However, like @Nathan-C described, you can stand up the required services using Azure Iaas (either DC+DirSync+ADFS or DC+Dircync w/pwd sync) in order to achieve single sign-on between your your Office365 apps and your on-prem apps. You would need to deploy a VPN link between Azure and your local network.
Azure AD is NOT "regular" Active Directory.
@MDMarra: Thanks for the hints, so I did:
The users from O365 can be exported by powershell using
Get-MsolUser | Select-Object City, Country, Department, DisplayName, Fax, FirstName, LastName, MobilePhone, Office, PasswordNeverExpires, PhoneNumber, PostalCode, SignInName, State, StreetAddress, Title, UserPrincipalName | Export-Csv C:\Temp\Azure_Export_2014_12_05.csv -Encoding UTF8
This exports all columns to CSV where I could find a mapping that looked appropriate. Those are not all columns, but many of them cannot be mapped to attributes in AD. Others, like the password, cannot be exported.
To import the users to AD, run in powershell
import-csv C:\Temp\Azure_Export_2014_12_05.csv -Encoding UTF8 | foreach-object {New-ADUser -Name ($_.Firstname + "." + $_.Lastname) -SamAccountName ($_.Firstname + "." + $_.Lastname) -GivenName $_.FirstName -Surname $_.LastName -City $_.City -Department $_.Department -DisplayName $_.DisplayName -Fax $_.Fax -MobilePhone $_.MobilePhone -Office $_.Office -PasswordNeverExpires ($_.PasswordNeverExpires -eq "True") -OfficePhone $_.PhoneNumber -PostalCode $_.PostalCode -EmailAddress $_.SignInName -State $_.State -StreetAddress $_.StreetAddress -Title $_.Title -UserPrincipalName $_.UserPrincipalName -AccountPassword (ConvertTo-SecureString -string "Secret!" -AsPlainText -force) -enabled $true }
This creates new users with the name Firstname.Lastname. Other attributes like SignInName could not be used because they are not a valid AD account name.
Country cannot be imported because AD requires the country to actually exist while O365 accepts free text.
The password will be set to "Secret!", because if no password is provided, the account will be created, but disabled.
It may be handy to edit the CSV-file in Excel or something, but I would recommend using PowerShell only. Excel deletes leading zeros from phone numbers or reformats other stuff. Also, mind UTF8.
Best Answer
I don't think there's any official deployment guides for AD on Azure - and yes, you would still need to deploy ADFS or DirSync, because your AD (even in a cloud) wouldn't magically trust Office 365's infrastructure.
Direct Access only works on Windows 7 or 8, Enterprise Edition or Ultimate (for 7.) No Mac client. Domain join via DA can be tricky although I understand it works better with Win2012 R2 now.
For shared storage, why don't you just use SkyDrive and/or SharePoint in Office 365? That's what it's there for.
Instead of AD, which really doesn't do this well yet, perhaps you can look into things like InTune or some other MDM (mobile device management) tool?